Conformal Minimal Foliations on Semi-Riemannian Lie Groups

Abstract

We study left-invariant foliations $\mathcal{F}$ on semi-Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations $\mathcal{F}$ when the subgroup $K$ is one of the important groups $\mathrm{SU}(2)$, $\mathrm{SL}_2(\mathbb{R})$, $\mathrm{SU}(2)\times\mathrm{SU}(2)$, $\mathrm{SU}(2)\times\mathrm{SL}_2(\mathbb{R})$, $\mathrm{SU}(2)\times\mathrm{SO}(2)$, $\mathrm{SL}_2(\mathbb{R})\times\mathrm{SO}(2)$. This way we construct new multi-dimensional families of Lie groups $G$ carrying such foliations in each case. These foliations $\mathcal{F}$ produce local complex-valued harmonic morphisms on the corresponding Lie group $G$. This means that they provide the existence of solutions to a difficult over-determined non-linear system of partial differential equations.